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Waterbed Effects and Buyer Mergers

by

Adrian Majumdar

ESRC Centre for Competition PolicyUniversity of East Anglia

and

RBB Economics

CCP Working Paper 05-7

Abstract: This paper demonstrates how a profitable, downstream merger can lowerthe merged entity’s input price while raising that of its rivals, leading to an adverseeffect on final consumers. This novel ‘waterbed’ result is surprising and very different to the unilateral and co-ordinated effects usually considered in the analysisof horizontal mergers. When demand is linear, all mergers involving a powerfulbuyer harm overall welfare even though the merger leads to marginal cost reductionsthat substantially increase output by the merged entity.June 2005

Keywords: Mergers; Buyer Power; Raising Rivals’ Costs.JEL Classification: L40, D43

Acknowledgments:I am grateful to Stephen Davies, Morten Hviid and Bruce Lyons for several helpfuldiscussions and comments which have substantially improved the paper. Remainingerrors are my own.

Contact details:adrian.majumdar@rbbecon.com

ISSN 1745-9648

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Waterbed effects, countervailing buyer power and buyer mergers.

1. Introduction

1.1 Waterbed effectsCan a merger of two buyers lower the merged entity’s input price while raising that of its rivals as suppliers make up their lost margins from weaker buyers? An economist’s first instinct is usually to answer: ‘No! If suppliers could raise prices elsewhere, why are they not already doing it?’

Nevertheless, this ‘waterbed’ effect has influenced policy makers intwo recentmergers. The United Kingdom Competition Commission (CC), during aninvestigation of several prospective supermarket mergers, states: ‘The exercise of buyer power by the merged entity would have adverse effects on other, smaller,grocery retailers through the “waterbed” effect - that is, suppliers having to chargemore to smaller customers if large retailers force through price reductions whichwould otherwise leave suppliers insufficiently profitable’ (CC (2003) paragraph2.218). The CC also considered that a similar effect might occur as a result of amerger in the private health care market (CC (2000) paragraph 2.180 b).

The motivation for this paper is to establish conditions in which a merger gives rise tosuch a waterbed effect and to examine the implications for welfare. To myknowledge, this is the first paper to address the issue.1 In related papers, Matthewsonand Winter (1996) and Gans and King (2002) demonstrate how the introduction ofbuyer power (modelled by a first mover advantage in dealing with suppliers) canbenefit the powerful buyer while leaving other buyers worse off. However, theseauthors do not address the specific issue of how pre-existing buyer power may beenhanced by a merger with adverse effects for final consumers in a downstreammarket (see section 5 for a further discussion).

I consider a setting where there is a single procurement market upstream and several‘local’ Cournot markets downstream. In the procurement market a powerful buyer, R,contracts directly with its suppliers. Then, other buyers purchase inputs on a spotmarket. I consider R’s incentive to purchase stores in local markets where it does not already have a presence. This allows me to concentrate on mergers that leave thelocal market structure unchanged. The merger is nevertheless a horizontal merger inthe sense that it is a merger of two buyers in the procurement market.

I find that a buyer merger typically reduces welfare. This is a novel theory of how amerger may harm consumers. Even though the merged firm increases outputcompared to pre-merger levels, output declines overall (i.e. aggregated across localmarkets) due to the higher input price that rivals face. This model therefore differssignificantly from the class of mergers analysed by Farrell and Shapiro (1990). Intheir paper, if a merger of Cournot firms generates synergies which induce the merged

1 This paper is concerned only with a waterbed effect arising from a merger of two single productfirms. The term waterbed effect has also been used in relation to multiproduct firms where price capsare imposed. Broadly speaking, the idea is that if a multiproduct firm is subject to a break evenconstraints (e.g. due to rate of return regulation or due to intense competition from other similar firms)then capping the price on one line of business may lead to a price rise on another line of business.

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entity to increase output (compared to the sum of the pre-merger outputs of themerging firms) this is sufficient for welfare to increase. With the waterbed effectdescribed in this paper, however, a rise in the merged entity’s output is not sufficient to improve welfare.

The waterbed effect arises from two features. First, the merger allows the target storeto benefit from the lower prices negotiated by the powerful buyer. Second, themerger reduces demand from independent stores that purchase in the spot market.The fall in their demand weakens the constraint provided by a potential entrant to thespot market, as that entrant would have to charge a higher price to recover its entrycosts.

The waterbed effect also provides a different angle on raising rivals’ costs. In thisliterature, harm to competition may arise where a dominant firm profits from raisingits own costs because this raises rivals’ costs by sufficiently more as set out in theseminal paper by Salop and Scheffman (1983). However, as Mason (2002) notes,cost raising strategies usually require strong restrictions on parameters if they are tobe profitable. This is because the direct effect on profits is negative (own costs go up)and this must be outweighed by a very strong indirect effect (i.e. that arising fromrivals having higher costs).2 In general, it is difficult to find ‘costless’ cost raising strategies. With the waterbed effect, however, raising rivals’ costs by merger is profitable because it is accompanied by a reduction in own costs. Even absent thecost raising effect, the merger would be profitable.

The rest of the paper is set out as follows. In section 2, we set out the model. Section3 then addresses welfare implications of a merger. Section 4 discusses the robustnessand realism of the key assumptions in the paper. Section 5 compares and contrasts theresults to previous literature on buyer power and buyer mergers. Finally, section 6offers concluding remarks and policy implications.

2. The model

2.1 AssumptionsThere are two identical manufacturers, M1 and M2. In order to produce, each suppliermust sink a fixed cost, F > 0. F is assumed to be sufficiently low to allow at least onemanufacturer profitably to supply buyers (the specific condition is set out below insection 2.2). Having sunk F, manufacturers can produce an identical input at aconstant marginal cost, equal to zero.

Manufacturers sell to retailers in a procurement market. Retailer R has a first moveradvantage in dealing with manufacturers which means that manufacturers establishR’s input price before they make offers to other retailers (timing assumptions are described in detail in the following section). R may own more than one storeprovided that it does not own two or more stores in the same local market. Otherretailers, which we alsorefer to as ‘independents’, own only one store each.

2 Mason (2002) focuses on the other strand in the raising rivals’ costs literature that firms might raise each other’s costs in order to dampen competition.

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There are m separate ‘local’ retail markets, where m 2. Each local marketpotentially has n identical retail stores, where n 2. Stores in each local market areCournot competitors. Each local market faces the same market inverse demand form,P(Q), where P’(Q) < 0 and P’’(Q) 0. All assumptions are common knowledge.

2.1.1 Timing assumptions

Stage 1–retail merger R, which begins the game owning one store, purchases stores in as many local

markets as it wishes provided that R does not own more than one store in anylocal market. We denote the number of stores owned by R at the end of stage 1 byr, where 1 r m.

Stage 2–invitation to tender R invites manufacturers to bid for the right to be its sole input supplier. Manufacturers place bids simultaneously by means of a sealed bid auction. Bids

specify the input price at which R may purchase whatever amount it desires. R appoints a manufacturer and that manufacturer commits to producing by sinking

F. R’s decision to appoint a manufacturer and that manufacturer’s commitment toproduction are public knowledge. R may decline both offers if neither would beprofitable.

Stage 3–spot market price setting Each manufacturer simultaneously announces a price at which it is willing to

supply the remaining retailers. We denote this the spot market price, si (i =1,2); si

is observed by all players.

Stage 4–procurement orders All retailers simultaneously and non-cooperatively place orders with

manufacturers. They incur a small, fixed transaction cost per order. Manufacturers supply spot market purchasers provided that their continuation

profits are non-negative. If a manufacturer’s continuation profits are negative, it withdraws its spot market offer. If one manufacturer withdraws its offer while theother does not, retailers that had ordered from the former are given the opportunityto purchase from the latter at the latter’s spot market price. If both manufacturerswithdraw offers, spot market buyers are not supplied the input.

Stage 5–retail competition Retailers transform their inputs into the final product at zero cost. One unit of

input is required to produce each unit of output. Retailers are Cournot competitors in each local market.

The robustness and realism of the key assumptions are discussed in section 4.

2.2 Results for stages 2 to 5.First we solve the game for stages 2 to 5, taking the number of stores owned by R asgiven (we consider stage 1, the incentive to merge, in section 2.3). We proceed by

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backwards induction. Rather than set out all definitions up front, we introduce themas we proceed so as to provide context to the particular variables defined.

Stage 5 –retail competitionAt stage 5, each local market is characterised by Cournot competition among activeretailers. We define the following outputs: qRj(c,s,aj), output by an active store owned by R in market j; qLRj(c,s,aj), output by an active independent store that competes in a local market

j, where R also has an active store; qLk(s,ak) output by an active independent store that competes in a local market k,

where R does not have an active store,

where c refers to R’s input price, s the spot market price, and aj and ak the number ofactive stores in a given local market (aj n, ak n). Output and profit at each storeare decreasing in the store’s input price and increasing in the input price paid by other stores that compete in the same local market.

Define QS(c,s,a) to be total output by independent stores and QR(c,s,a) to be R’s totaloutput, where a stands for the relevant ‘aj’ and ‘ak’ variables. (It turns out that allfirms will be active in equilibrium.)

Since each local Cournot market is characterised by constant marginal costs and(quasi) concave demand, this is sufficient for the stage 5 equilibrium to be unique.3

Stage 4- procurement ordersWe now consider orders placed with manufacturers. Given that R does not declineboth offers at stage 2, we may assume without loss of generality that R chooses todeal with M1. At stage 4, therefore, M1 will have already sunk F, and so itscontinuation profit from supplying spot market purchasers is non-negative providedits spot market offer, s1 is also non-negative.

On the other hand, M2 has not yet sunk F. Thus its continuation profit is positive ifand only if:

[1] s QS(c,s,a) F,

where s QS(c,s,a) is the total revenue available from spot market sales and is M2’s share of those sales. Equation [1] is M2’s participation condition.

If manufacturers announce different (non-negative) prices, all spot market buyerspurchase from the lowest priced manufacturer provided that this would allow thatmanufacturer to make non-negative continuation profits. More formally, denote M2’s spot market offer s2. If s1 < s2, spot market purchasers will place orders only with M1

(provided that they expect M1 to meet its participation condition, i.e. provided s1

3 See Farrell and Shapiro (1990) equations (3) and (4).

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04). Likewise, if s2 < s1, spot market purchasers place orders only with M2 (i.e. = 1),given that such orders are expected to allow M2 to meet its participation condition.5

An interesting issue is what happens if manufacturers announce the same spot marketprice, i.e. s1 = s2. Consider one possibility which we will make use of in ourdiscussion of stage 3. Suppose that the only way for M2 to meet its participationcondition is to supply all spot market purchasers. In this case, all it takes is for onebuyer to purchase from M1 in order for M2 to withdraw its offer. As a result, buyersthat originally ordered from M2 would still have the opportunity to purchase at prices2 (since they can purchase from M1 at s1 and s1 = s2), but they would incur a second(small) transaction cost of ordering from M1. Thus, spot market buyers would have aweakly dominant strategy of placing orders with M1 in the first instance as thisensures they pay the transaction cost only once.6

Stage 3 –spot market price settingAt stage 3, spot market prices are announced simultaneously. To keep stage 3interesting, we make two minor assumptions. First assume that spot market demandis inelastic over the range (0,smax), where smax ≡ arg max s QS(c,s,a). In other words,smax is the spot market price which maximises spot market revenue. Second, assumethat the fixed cost, F, is sufficiently low so that a spot market price always existswhich is less than smax and which generates spot market revenue at least as high as F.

Consider the lowest possible price that M2 could announce that is a candidate formeeting its participation condition [1]. This price, sb, is the lowest value for s thatsatisfies the following condition:7

[2] s QS(c,s,a) = F

Suppose M2 offers sb and consider M1’s possible best responses. Since sb > 0, M1’s best response is to offer the same price.8 Buyers will then purchase from M1 as this istheir weakly dominant strategy, as explained in the previous subsection. Offering a

4 It would not be rational for a retailer to order from a manufacturer that was not expected to deliver onthat order since that retailer would incur a transaction cost without expecting to obtain supplies.5 This statement is not quite as straightforward as it may first appear. Suppose that the only way for M2

to meet its participation condition is to supply all spot market purchasers. In this case, even though s1

> s2, a buyer might conceivably purchase from M1 and thereby ensure that any active retailer thatpurchases on the spot market must also pay the higher input price, s1. This might be profitable if somebuyers were expected to order from M2 and, as a result of M2 withdrawing its offer, these buyers wouldremain inactive rather than incur the second transaction cost required to order from M1. We rule outthis possibility by assuming that the fixed transaction cost incurred per order is very small. We alsoassume that if all buyers pay the same input price, their profits are decreasing in that price. Thisensures that if s1 > s2 a buyer would not find it profitable to purchase from M1 to ensure that all otherspot market buyers paid the higher price as well.6 The assumption that buyers that originally order from M2 would have the opportunity (for a smalltransaction cost) to purchase at price s1 if M2 withdraws its offer is not essential. Instead, we mightnote that if s1 = s2 = sb, buyers naturally purchase from M1 as to purchase from M2 would not berational if there is a small chance that one buyer ‘trembles’ and erroneously purchases from M1.7 Note that, for sufficiently low values of F, there will typically be two input prices that satisfy thiscondition. If so, only the lowest price is relevant. If one manufacturer charged a higher price thatsatisfied [2], the other could profitably undercut.8 Provided that c 0. If c < 0 (which will not occur in equilibrium), M1 may have an incentive to bidbelow sb to shift sales away from R where it makes a loss in the continuation game.

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higher price would forfeit profitable sales to M2 while offering a lower price wouldnot be as profitable as offering sb.

What price will M2 offer? If it offers any prices’in the range sb <s’< smax, M1’s best response is to offer s’–, where is an arbitrarily small amount. However, if M1

were to offer s’–, M2 would find it profitable to undercut for any s’–> sb.Therefore, equilibrium occurs only when M1 and M2 both announce a price of sb.

It follows that in equilibrium all spot market buyers purchase from M1 and M1 earns acontinuation profit equal to F. M2 receives no orders and earns its outside option,zero.

Stage 2 –invitation to tenderOur discussion of stages 3 and 4 demonstrates that R’s supplier is assuredof coveringits fixed costs while the other supplier earns zero. Stage 2 is thereforestraightforward. Intense competition among suppliers (resulting from the sealed bidauction to be R’s only supplier) induces both suppliers to offer an input price, c, equalto zero. R then arbitrarily chooses a supplier. For convenience, we have adopted theconvention that R deals with M1.

Even though M1 makes no revenues from sales to R, it is rational to sink F duringstage 2 because M1 knows that it will recover its fixed cost at stage 4 and break evenoverall.

Interim summarySo far we have shown that R deals with M1 and obtains its input at a price zero. M1

sells to all remaining stores who purchase on the spot market at sb. It follows that allstores are active in the downstream market (i.e. in equilibrium, aj = ak = n for all j andk).9

2.3 Stage 1–incentives to merge and the waterbed effectWe now turn to stage 1 and consider R’s incentives to merge(i.e. we allow r to vary).First we establish a lemma that any merger leads to an increase in the spot marketprice. Second we establish a lemma that successive mergers are profitable. Finally,we establish our first proposition–the existence of a waterbed effect.

Lemma 1: As the number of stores owned by R increases, the spot market price goesup.

Before proving this lemma, it is helpful to define the following terms. Consider storelevel demand. In any of the r markets where R operates, output by each of R’s stores is the same; we denote this qR(sb(r)).10 In each of the markets where R sells, therewill be (n–1) identical independent stores; we denotean individual store’s output as

9 Since n is finite, the Cournot equilibrium is such that each firm receives a price above its unit inputcost. Thus R and the other stores all make positive profits which by assumption exceed the only fixedcost that they face (i.e. the small transaction cost of dealing with M1).10 Our previous discussion solved stages 2 to 5 of the game for a given value of r in the range 1 r m.We now introduce r as an argument to our solution prices and quantities (we drop c as an argumentsince c = 0 in equilibrium).

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qLR(sb(r)). In each of the remaining (m –r) markets, there will be n identicalindependent stores; we denote an individual store’s output as qL(sb(r)).

It follows that spot market demand is:

[3] QS(sb(r)) = (m–r) n qL(sb(r)) + r (n–1) qLR(sb(r))

We now prove that the derivative of sb(r) with respect to r is positive in the range 1 r m.

Proof. From [2] and [3]:

[4] sb(r) {(m–r) n qL(sb(r)) + r (n–1) qLR(sb(r))} –F = 0

Denote equation [4] as f(sb(r)). By the implicit function rule:

[5] dsb(r)/dr =–f(sb(r))/r / f(sb(r))/sb(r).

The numerator of [5] equals:

[6] –sb(r) {–n qL(sb(r)) + (n–1) qLR(sb(r))}

and is positive since qL(sb(r)) > qLR(sb(r)).

The denominator of [5] equals:

[7] (1–(sb(r))) QS(sb(r)),

where (sb(r)) is the absolute value of the elasticity of spot market demand evaluatedat sb(r). We made the minor assumption that 0 < (sb(r)) < 1 (see the discussion ofstage 3 above and note that this assumption is not required if local demand is linear)and hence the denominator is positive. Thus the sign of [5] is positive. QED.

There is an intuitive explanation for why sb(r) increases with r. Recall that the spotmarket price must generate revenues equal to the fixed cost, F. As R owns morestores there are fewer independent stores over which F can be spread out. Thus, eachindependent store must pay a higher input price. Moreover, as independent stores paya higher price, their input demands decline. Thus, F is recovered not only over fewerstores but also each store purchases a smaller quantity. Both effects work to push upsb(r).

Lemma 2: Each successive merger is increasingly profitable for R (up to the pointwhere further mergers are not permitted).

Proof. Consider what happens when R purchases one extra store (a ‘target’ store) in a ‘target’ market. By purchasing the target store, that store’s input price falls from the spot market price to zero and so, in this sense, R’s input price falls in the target market. R’s input price in all the other stores it owns remains unchanged (at zero).

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Profit earned at the target store increasessince the store’s input price falls while, from lemma 1, the input price of its rivals goes up. Profit earned at R’s other stores also increases since they compete with higher cost rivals.

All of R’s stores earn identical profits. Since the profit of a store that R alreadyincreases as R purchases another store, it must be the case that R’s profits are increasing in r. This means that at stage 1, R purchases stores in every market whereit did not previously have a store. In other words, R purchases m–1 stores in order tohave a presence in all local markets and so stage 2 commences with r = m. QED.

Proposition 1: A ‘waterbed effect’exists. As R extends its buyer power by aprofitable merger (here modelled by an increase in r in the range 1 r m), R’s input price falls in the target market and remains unchanged elsewhere, while itsrivals’ input price increases.

Proof. This follows from lemmas 1 and 2 above.

3. Welfare effects of the merger

In this section we consider how a merger affects overall welfare which we define asthe sum of consumer and producer surplus. We denote overall welfare as W(r). Firstwe demonstrate that with linear demand mergers always harm overall welfare, eventhough the output of the merged firm increases substantially. Second we test therobustness of this finding to an alternative demand specification. Finally, we explainthe novelty of our result and compare our findings with the seminal paper on Cournotmergers by Farrell and Shapiro (1990).

3.1 Linear demandOverall welfare is the sum of welfare in each local market. The contribution tooverall welfare from any particular local market increases with output in thatmarket.11 When R purchases a store in a target market, the change in total welfare canbe broken down into the three following effects. First, output must fall in any localmarket where R has no presence (since sb(r) rises). Second, output must fall in anymarket where R already had a presence (since sb(r) rises while R’s input price remains unchanged). Both these effects are bad for welfare.

However, the third effect–in the target market–is ambiguous. The target store facesa lower input price while other stores have a higher input price. Therefore, if theraising rivals’ cost effect is relatively weak, output may increase in the target market.

It turns out that with linear demand, any increase in welfare in the target market doesnot offset the decrease in welfare in other markets. This is our second proposition.

11 Farrell and Shapiro (1990) note that in a Cournot market welfare may increase even though outputdeclines due to output being shifted from high cost firms to low cost firms. However, in our model thisdoes not apply. The marginal cost of interest for overall welfare is M1’s marginal cost of production which is zero. Therefore, welfare in any local market is simply the area under the demand curvebetween the origin and the level of local output.

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Proposition 2: When local demand is linear, welfare always declines as a result ofthe merger (i.e. dW(r)/dr < 0).

Proof. See appendix.

This is a striking result. For a commonly used demand form, any merger is harmfuleven though it generates efficiencies for the merged entity (i.e. a substantial fall inmarginal cost for R’s store in the target market). We discuss this further below.

3.2 Concave demandIn this section we discuss welfare effects more generally.

We explain in the appendix that if the merger leads to a rise in the sum of marginalcosts in the target market, this is sufficient (but not necessary) for overall welfare tofall. Formally, if [8] holds, overall welfare must fall because output falls in allmarkets, including the target market (see lemma 3).

[8] (n –1) dsb(r)/dr > sb(r)

In other words, if the raising rivals’ cost effect is strong enough, the increase in the input price paid by independent stores will be so great that output will fall in thatmarket despite the fall in R’s costs.

Further, we show in the appendix that the more stores that R owns in the first place,the higher the likelihood that purchasing one more store will harm overall welfare(see lemma 4). Intuitively, if F is spread over several independents then the loss ofone independent is hardly noticeable. However, if F is spread over relatively fewindependents, then the loss of one more independent has a marked effect on the rest.So, the more stores that R owns (and hence the fewer independents there are), thegreater is theraising rivals’ cost effect and the more likely that a merger will lead to areduction in output in the target market.

Interestingly, from lemma 2 we know that the incremental profit from purchasing anadditional store is also increasing in r. Therefore, R’s strongest incentive to merge occurs when r is ‘high’, that is when the merger has the greatest harm (or smallest benefit) for welfare.

Finally, we consider the following form of concave demand as a robustness check.

[9] P(s(r)) = a –b Q(s(r)) k

We show that, with this form of demand, welfare is higher when r equals zero (i.e. allpurchases are made on the spot market), than when r equals m (i.e. R has a presencein all local markets). It follows that even if a merger increases welfare to start offwith, there must come a point when the merger starts to harm welfare. Once we reachthis point, welfare falls for any further mergers (see proposition 3). Our suspicion isthat welfare declines with any merger although we have not proved this.

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3.3 A novel resultOur model provides an important result. For many possible forms of (quasi) concavelocal demand, a profitable merger that induces the merged entity to increase outputmay reduce welfare. At best, when R purchases an additional store the mergerbenefits consumers only in the target market. Consumers in other markets pay higherprices. At worst, when [8] holds (which is more likely when R already owns manystores), all consumers pay higher prices.

The latter finding is particularly interesting because it departs significantly from thetraditional view of horizontal mergers. The analysis of horizontal mergers is usuallyconcerned with ‘unilateral’ (or equivalently ‘non-coordinated’) effects and co-ordinated effects. In the former case, the potential concern is that the reduction inconcentration will provide the merged entity with an incentive to restrict output.Unless efficiencies are particularly strong, the merger moves the market to a higherpriced equilibrium. In the latter case, the concern is that the merger makes tacitcollusion more likely or strengthens existing tacit collusion.

In our paper, the merger generates efficiencies and induces the merged firm toincrease output to a level that exceeds the sum of the pre merger output of itscomponent stores. In the Cournot models considered in the seminal paper by Farrelland Shapiro (1990), this would be sufficient to increase welfare in the target market.However, unlike those authors, we also have a raising rivals’ cost effect due to thereduction of spot market demand. This is sufficient to reduce welfare in the targetmarket when [8] holds.

In sum, our harm to consumers arises solely from a buyer power effect in theprocurement market. The merger would be profitable even absent the raising rival’s costs effect but, because of that effect, welfare is harmed. The harm to consumersdoes not arise from unilateral or co-ordinated effects because we leave both the localmarket structure and the form of competitive interaction unchanged. This, to theauthor’s knowledge, is a new theory of harm from a horizontal merger.

4. Discussion of key assumptions

In this section, we discuss the robustness of the model’s key assumptions and inwhich situations the model may have practical relevance. We demonstrate that theintuitions from the model apply more widely than in the specific setting described inthe preceding sections.

At stage 2 (invitation to tender), R derives its buyer power from two features. First, ithas the ability to have suppliers compete at a sealed bid auction (where the twosuppliers have full information and common valuation) for the right to be its soleinput supplier. Costs of making a bid are assumed to be zero. Under theseassumptions we have a standard Bertrand competition among identical suppliers ofhomogenous goods.

Our stage 2 results are most likely to be relevant where suppliers are similar and haveboth agood knowledge of each other’s costs and a reasonable idea of the value of the

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‘prize’ (i.e. the continuation profit available in the spot market).12 Interestingly, thissuggests that stage 2 of the model is less likely to apply to branded goods (in contrastto the view put forward by CC (2003)) and perhaps more likely to apply to suppliersof own label products.13

The second feature at the invitation to tender stage is R’s first mover advantage; Rdeals with suppliers before other buyers.14 This begs the question: why might R havefirst mover advantage? We might justify R’s first mover advantage by assuming thatR already is large in relation to the purchasing market and that only R can profitablycontract outside the spot market due, say, to economies of scale in striking thesecontracts.15

R could be large in relation to the purchasing market for one or more of the followingreasons: R owns many stores instead of just one store. (We assumed that R owned only

one store at the start of the game; however, if R owned more than one store theresults are unchanged provided there remains scope for R to purchase more stores,i.e. r < m.);

R purchases the input for use in another area (where it does not compete withindependents) and so already has dealings with the two suppliers;16

in the downstream market, there is an additional, constant per unit cost oftransforming inputs to outputs and that R has a significantly lower marginal costof transformation than other stores;

R and M1 are vertically integrated.

Any of the above assumptions can be added to the model without qualitativelyaffecting the results.

A chain store that purchases on a national market but sells into a local market may fitsome of the above possibilities. (Note that the model need not be restricted to a singlechain. In principle, other chain stores could exist without necessarily affecting thequalitative results of the model.17)

12 Many procurement markets have a bidding process, although not all would necessarily satisfy theconditions for Bertrand competition (see Klemperer (2005)).13 An alternative interpretation of stage 2 is as follows. Suppose that prior to the game M1 and M2

already supply the procurement market but at a very high constant marginal cost. F could be theexpenditure required to make a major innovation (e.g. an investment in a state of the art distributionsystem) that substantially reduces the marginal cost of supplying that procurement market (here, tozero).14R’s first mover advantage is not essential. Suppliers could simultaneously announce prices to R andto the spot market without affecting the results, provided that R announces with whom it will deal andthat supplier commits to producing before purchase decisions are made (this ensures that all spotmarket buyers purchase from the same supplier). In this case it is R’s ability to deal with suppliers on an individual basis that matters.15 Stages 2 and 3 of our model are similar to the game modelled in Gans and King (2002). The latterassume that small buyers are not sufficiently large to contract with.16 For example, suppose M1 and M2 produce an identical own label product and also (potentially)supply the own label product to the spot market. These suppliers may also produce a branded productwhich they distribute through R (and so R would already have dealings with these suppliers).17 For example, there could be other powerful chain stores that compete in the downstream market buthave alternative suppliers that are contracted not to supply other buyers (and so cannot supply the spotmarket). They would not directly be affected by the change to the procurement market (but they may

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The fundamental assumption that drives this model is that reducing spot marketdemand leads to a higher spot price being established at stage 3. In this model, takinga potential buyer out of the spot market means that fixed costs must be recovered overfewer sales and so the spot market prices rise. This effect is driven by what amountsto a contestable spot market. Although M1 has already sunk the fixed cost required toproduce, if M1 attempts to earn continuation profits that exceed F at stage 3, M2 canundercut by offering a break even price, sb(r), and thereby win all spot market buyers.However, as R purchases more stores, the pool of buyers available for M2 declinesand so sb(r) increases–M2 has fewer buyers over which to spread its fixed costs.18

Given the Bertrand competition at stage 2 (invitation to tender), it is reasonable toassume a ‘contestable’ spot market.That is, having assumed similarity of suppliers,common knowledge of costs and absence of costs of making a bid at the tenderingstage with R, the Bertrand spot market is the natural way to model competition for theremaining buyers (which, by assumption, are too small to contract with on anindividual basis).

We could generate the waterbed effect in alternative ways. For example, if averagecosts decline with output up to a point, similar qualitative results can be derived.19

Intuitively, where the merger reduces spot market demand sufficiently to prevent M2

supplying the spot market at an efficient scale, M2 becomes a weaker constraint on M1

and a waterbed effect arises which may reduce welfare.

Finally, we have assumed that supplier-retailer contracts specify an input price only.At stage 2 (invitation to tender) nothing turns on this. If M1 were to offer acombination of a fixed fee and a price at which R could purchase unlimited quantitiesof the input, the best offer it could make is, in effect, the same. M1 would not requirea fixed payment and would offer the input at marginal cost (i.e. zero).20

What is important (but not essential21), however, is that at stage 3 the spot marketspecifies only the input price. This ensures that as spot market demand shrinks, F isrecovered by a higher spot price as opposed to higher fixed fees for spot marketbuyers. This can be thought of as a stylised representation of supplier-buyer deals

gain indirectly to the extent that independent stores face higher prices in the local markets where theycompete). Alternatively, there could be chain stores that buy on the spot market but which are notlarge enough to strike deals with suppliers outside of the spot market.18 In principle, M1 does not need to sink F at stage 2 (invitation to tender) if its commitment toproduction can be made in another way (e.g. through a penalty clause that if it fails to supply R then itmust pay a cost in excess of F.19 This type of cost function was popularised in Rasumsen, Ramseyer and Wiley (1991) and used, forexample, in Gans and King (2002).20 We would not expect M1 to offer an input price below zero in return for a fixed fee. This is becausein the continuation game, M1 would have the incentive to lower the spot market price below sb toreduce its loss making sales made to R.21 If we allowed for fixed fees then we might still obtain a reduction in welfare as follows. Supposethat spot market buyers reimburse M1 by means of a fixed fee. Suppose also that spot market buyersare identical other than that they face an additional fixed cost of production in the downstream marketwhich varies among individual stores. As the spot market payment increases (i.e. the fixed fee goesup), this could lead some individual stores to exit the downstream market. The higher concentrationwould then lead to consumers paying higher prices in some local markets.

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where, even if fixed transfers take place, the input price is by far the key parameter indeterminingeach party’s payment.

5. Discussion of related literature

5.1 Buyer powerRegarding the modelling of buyer power, Gans and King (2002) and, to a lesserdegree, Matthewson and Winter (1996) are the papers most closely related to this one.Both papers show how the buyer’s first mover advantage allows the buyer to obtainbetter terms than the remaining buyers.

The stage 2 and 3 timing assumptions in this paper (invitation to tender and spotmarket pricing) draw closely on Gans and King (2002). These authors consider howexclusionary contracts between a large buyer (or a group of large buyers acting asone) and a supplier can be socially sub-optimal. In their model, the large buyer has afirst mover advantage in striking a two-part tariff contract with one or both of twoidentical suppliers. Having dealt with the large buyer, suppliers then make offers to‘small’ buyers on a spot market.

Gans and King (2002) assume that suppliers have average costs that fall with outputup to a critical point Q* after which they are constant and equal to c(Q*). They alsoassume that demand from the spot market is insufficient to allow a supplier to sell Q*units at a price of c(Q*) and that the large buyer purchases sufficiently more than Q*units when its input price is c(Q*). In this case, if the large buyer deals exclusivelywith one supplier there are two effects. First, the ‘contracting’ supplier achieves its efficient scale. Second, the other supplier is a weaker force in the spot market since itcannot achieve its efficient scale. This raises the equilibrium spot market price andharms welfare.22

Matthewson and Winter (1996) put forward the idea that providing a buyer with firstmover advantage can both lower that buyer’s costs and raise prices for other buyers.In their model, an ‘inside’ group of buyers is able to form a coalition which makes atake-it-or-leave-it offer to a subset of the population of suppliers. Only this subset ofsuppliers is active in the next stage, where there is monopolistic competition to supplyoutsiders (i.e. those not in the buying group). Suppliers are symmetric in that they allcharge the same price to outsiders. By restricting the number of active suppliers, thebuying group weakens competition to supply outsiders and thereby raises the pricethat outsiders pay. As a result of outsiders paying more, insiders are able to demand alower price while allowing suppliers to break even. Matthewson and Winter (1996)note that the buyer group’s decision to restrict the number of suppliers they deal with may be good for welfare where it reduces excess entry.

The key difference between our paper and those of Gans and King (2002) andMatthewson and Winter (1996) is that in their papers buyers are, in effect, finalconsumers. In contrast, we model buyers specifically as downstream firms andconsider how buyer mergers affect welfare in each local market as well as the whole

22 This is in contrast to the case where the large buyer has no first mover advantage and purchases onthe spot market. This would allow both suppliers to produce at efficient levels and so all buyers wouldbe supplied at c(Q*) and welfare would be higher.

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industry. Buyer demands are therefore interdependent, being linked through theprocurement market.

5.2 Buyer mergersChen (2003) considers the incentive for a retailer to enter one or more local markets.To set the scene, Chen (2003) assumes that a monopoly supplier sells to a dominantdownstream firm, D, and a downstream fringe. The supplier first announces a pricefor the fringe and then strikes an efficient bargain with D, taking the fringe price andD’s bargaining power as given. Finally, D and fringe firms compete to sell to finalconsumers.

The key to the model is D’s bargaining power with the monopoly supplier. Chenassumes that D takes a share, , of the surplus generated by its deal with themonopolist. The monopolist earns profit from two sources –its bargain with D andits sales to fringe firms. As increases, the monopolist’s share of the spoils from dealing with D declines and so, at the margin, sales to fringe firms become moreimportant. This induces the monopolist to lower the input price it offers fringe firms.

Prior to the game there are several downstream markets consisting only of fringefirms. At the start of the game, D can choose how many markets to enter (this issimilar to our buyer merger stage) and is increasing with the number of markets inwhich D is active (e.g. because this makes backwards integration more profitable).This means that D faces a trade off: as increases D obtains a larger share of thebargaining pie but faces greater competition from the fringe (which reduces the sizeof the bargaining pie). As a result, D may limit the number of markets in which it isactive to ensure that its bargaining power is not too high.

The results in Chen (2003) results are almost directly opposite to those in our paper:the ‘merger’ (i.e. taking a presence in a local market where previously there were fringe players only) lowers rivals’ costsand buyer ‘mergers’may be profitable onlyup to a point (whereas in our paper mergers become more and more profitable as rincreases). One key difference in the two papers is the nature of upstreamcompetition. In Chen (2003) there is no upstream competition, while in our paperupstream competition is intense.

This begs the question: what happens with intermediate forms of upstreamcompetition in our model? With weaker forms of upstream competition, the nature ofthe cost curves would be the key to deriving a waterbed result.23

23 For example, suppose that M1 is the only supplier. Assume that, at stage 2, M1 strikes an efficientbargain with R (in which R is supplied at marginal cost and a fixed fee is paid depending on relativebargaining strengths). Suppose that the merger enhances R’s bargaining power with M1 (e.g. because itmakes self supply more credible). Suppose also that dealing with R adds incremental surplus (i.e. M1’s fall back option of selling only through independent stores is worse than when it sells through R aswell) and that R cannot contract with M1 to prevent a spot market price being set at stage 3. Underthese conditions, the merger may lead to a fall in spot market demand and a higher spot market pricewhere M1’s marginal cost rises sufficiently steeply as output declines. Thus, even without upstreamcompetition, a waterbed effect could occur.

16

Our paper also differs from a strand of recent contributions that address how mergersamong buyers may be profitable through affecting the merged entity’s share of the incremental surplus resulting from an efficient bargain with a monopoly supplier. Forexample, Chipty and Snyder (1999), Inderst and Wey (2003), and Raskovich (2003)focus on buyers that operate in independent markets (so there is no incentive to mergein order to enhance downstream market power) and show that the profitability of amerger dependson whether the ‘gross surplus function’ is concave or not.24

Dobson and Waterson (1997) take a different approach to modelling buyer power andbuyer mergers. In their model, a monopoly supplier sells to symmetric, differentiatedretailers in a single downstream market. A retailer’s input price is determined by a Nash bargain struck with the supplier (there are no fixed fees). An increase in buyerpower is modelled by a reduction in the number of retailers (i.e. a buyer merger‘destroys’ a buyer). This typically reduces the supplier’s bargaining power with a retailer by reducing its fall back option (i.e. the profit available from selling to otherretailers). However, even though a decline in the number of retailers lowers the inputprice, it also raises market power in the retail market. Thus, unless downstreamcompetition is very intense (almost perfect), buyer mergers harm welfare because themarket power effect dominates and consumers pay more.25

In our setting, buyer mergers may also harm welfare, although for a different reason–their effect in the procurement market. Our model differs from Dobson and Waterson(1997) in three key respects. First, the merger does not ‘destroy’ a retailer so that the total number of stores remains unchanged after the merger–this allows us to model abuyer merger that does not directly increase downstream seller power. Second, whena merger destroys a firm, this begs the question of whether the merger would beprofitable. In our paper, mergers are always profitable (Dobson and Waterson (1997)do not address this issue). Third, we have competing suppliers upstream.

6. Concluding remarks - policy implications

We have demonstrated that a profitable downstream merger may lower the mergedentity’s input price, raise that of its rivals and leave consumers worse off. Thiswaterbed effect is very different from the unilateral and co-ordinated effects usuallyconsidered in horizontal merger analysis. The harm arises from a buyer power effectas opposed to a direct increase in seller market power.

Furthermore, we have proved the striking result that with linear demand all mergersinvolving a powerful buyer harm overall welfare even though the merger leads toefficiencies that substantially increase output by the merged entity. With moregeneral concave demand, mergers harm welfare after a point. The more profitable the

24 The intuition is as follows. Suppose that a monopoly supplier, M, deals with Z identical buyerssimultaneously. Each bargain struck is efficient and, taking other deals as being agreed at theirequilibrium values, maximises the extra surplus generated by striking the deal. Denote the latter‘incremental’ surplus, S1. S1 is shared between M and the relevant buyer according to some fixed rule(e.g. they get half each). Now suppose that z Z buyers merge. Denote the incremental surplusavailable from M’s deal with the merged entity as Sz. Broadly speaking, a concave surplus functionmeans that Sz/z > S1. Thus it is more profitable for z buyers to merge than to remain independent (for agiven incremental surplus sharing rule).25 von Ungern-Sternberg (1996) obtains a similar result.

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merger, the more likely it is to harm to welfare (and our suspicion is that mostmergers harm welfare in this setting).

In policy debates, the concern with buyer power created by mergers is often thatlower prices will not be passed on due to the simultaneous creation of seller power(e.g. Dobson and Waterson (1997)) or that buyer power would adversely affectupstream incentives to invest (e.g. CC (2003)). In each case, a case by case analysisis required as buyer power may have beneficial or anti-competitive effects.26 Thispaper presents a different potential concern that a buyer merger that is pursued inorder to pursue input price reductions can have an adverse side effect of raisingrival’s costs.

Our paper was motivated by the use of the waterbed theory in two recent mergerassessments in the UK where the argument was put forward that, as a powerful buyersecures a better deal from its suppliers, these suppliers make up lost profits from weakbuyers by charging them more.27 This paper provides a theoretical grounding for theassertion that a buyer merger could lower the buyer’s input price and lead to other buyers paying more. However, the mechanism is not that a supplier makes up lostprofits from weaker buyers. Rather, the buyer merger enhances the supplier’s market power over weaker buyers.

The waterbed result was driven by the feature that when a buyer is taken out of thespot market, demand falls and this leads to higher spot prices. We provided a specificmodel which demonstrates this effect and how welfare may decline as a result. Wethen explained that this result is not highly sensitive to most assumptions. The keyassumption is that average and/or marginal costs increase sufficiently as outputdeclines in the spot market.

The waterbed effect is therefore a theoretical possibility that should not be dismissed.Where it does exist, mergers are particularly profitable and the resulting welfareeffects may well be negative.

In terms of policy, there are two clear messages. First, before citing waterbed effectsas a reason for blocking a merger, competition authorities should specify a coherentmodel, supported by the facts, which explains how a buyer merger gives rise to asituation where suppliers are willing and able to charge other buyers a higher pricethan prior to the merger. In short, authorities must address the question: if supplierscan charge higher prices to the weaker buyers after the merger, why were they notcharging these prices before the merger?28

26 See Dobson and Waterson (1999). Note also recent contributions by Inderst and Wey (2002) whoshow that innovation may increase as a result of buyer power while Inderst and Shaffer (2004) providea model in which retail mergers may lead to single sourcing which adversely affects product variety.27 In a UK context, the CC included this type of waterbed theory in its report into prospective mergersin the private health care market and the supermarket sector (see CC (2000), CC (2003) and section 1.1above). In the latter case, the balance of survey evidence suggested that waterbed effects were unlikelyto occur, although a significant minority of responses thought otherwise, (CC, 2003). In a Europeancontext, waterbed effects have been recognised too (see for example, (Faull and Nikpay, 1999,paragraph 6.325).28 The CC did not address this question in CC (2000) and CC (2003).

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Second, there are risks involved when a buyer merger in an intermediate market isconsidered under separate merger regimes. For example, suppose that there arenational downstream markets and an international procurement market. To give thenational authority scope to address the downstream issues and then give the EuropeanCommission scope to address the procurement issues could make it less likely that theinteractions between the two levels are fully taken into account.29

29 In the supermarket retail merger Carrefour/Promodes, the European Commission (2000) referred theanalysis of the effect of the merger to the respective national authorities (France and Spain). This is notto argue that the waterbed effect had any relevance in this merger. Rather this is an example of howseparate merger regimes may consider different aspects of a merger.

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References

Chen, Z. (2003),‘Dominant Retailers and the Countervailing-Power Hypothesis’,Rand Journal of Economics, 34(4), 612-625.

Chipty, T. and C. M. Snyder (1999), ‘The Role of Firm Size in Bilateral Bargaining:A Study of the Cable Television Industry’, The Review of Economics and Statistics,81(2): 326-340.

Competition Commission (2000), British United Provident Association Limited andCommunity Hospitals Group plc: A report on the proposed merger; and BritishUnited Provident Association Limited, Salomon International LLC and CommunityHospitals Group plc; and Salomon International LLC and Community HospitalsGroup plc: A report on the existing mergers, Cm 5003, HMSO.

Competition Commission (2003), Safeway plc and Asda Group Limited (owned byWal-Mart Stores Inc); Wm Morrison Supermarkets PLC; J Sainsbury plc; and Tescoplc, Cm5950, HMSO.

Dobson, P. W. and M. Waterson (1997), ‘Countervailing Power and Consumer Prices’, Economic Journal 107, 418-430.

Dobson, P. W. and M. Waterson (1999), ‘Retailer Power: Recent Developments and Policy Implications, Economic Policy, 28, 133-164.

European Commission (2000), Commission Decision COMP/M.1664,Carrefour/Promodes, O. J. (C 164) 5.

Farrell, J. and C. Shapiro (1990), ‘Horizontal Mergers: An Equilibrium Analysis’, American Economic Review, 80, 107-26.

Faull, J. and A. Nikpay, eds, (1999), The EC Law of Competition, Oxford UniversityPress, Oxford.

Gans, J. S. and S. P. King (2002), ‘Exclusionary Contracts and Competition for LargeBuyers’,International Journal of Industrial Organization, 20, 1363-1381.

Inderst, R. and G. Shaffer (2004), ‘Retail Mergers, Buyer Power and Product Variety, mimeograph.

Inderst, R. and C. Wey (2003), Buyer Power and Supplier Incentives, CEPRDiscussion Paper No. 3547, London.

Inderst, R. and C. Wey (2003), ‘Bargaining, Mergers, and Technology Choice in Bilaterally Oligopolistic Industries’, Rand Journal of Economics, 34(1), 1-19.

Klemperer, P. (2005), ‘Bidding markets’, available at http://www.competition-commission.org.uk/our_role/analysis/bidding_markets.pdf

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Mason, R. (2002), ‘Cost-Raising Strategies in a Symmetric, Dynamic Duopoly’, TheJournal of Industrial Economics, 50(3), 317-335.

Matthewson, F. and R. A. Winter (1996), ‘Buyer Groups’, International Journal ofIndustrial Organization, 15, 137-164.

Raskovich, R. (2003), ‘Pivotal Buyers and Bargaining Position’, The Journal ofIndustrial Economics, 51(4), 405-426.

Rasmusen, E., J. Ramseyer, and J. Wiley (1991), ‘Naked Exclusion’, AmericanEconomic Review, 81, 1137-1145.

Salop S. C. and D. T. Scheffman (1983), ‘Raising rivals’ costs’, American EconomicReview, 73(5), PP 267-271.

von Ungern-Sternberg, T. (1996), ‘Countervailing Power Revisited, InternationalJournal of Industrial Organization, 14, 507-520.

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Appendix

Proof of Proposition 2: When local demand is linear, welfare always declines as aresult of the merger.

Suppose in any local market, j, inverse demand is:

[A1] Pj = a –b Qj(s)

Note that s is a function of r. However, to avoid notational clutter, we refer to sinstead of s(r). [A1] gives rise to the following Cournot outputs for independents andR:

[A2] qL(s) = (a–s) / (n + 1) b

[A3] qLR(s) = (a –2s) / (n + 1) b

[A4] qR(s) = (a + (n–1)s) / (n + 1) b

where, as in the main text, qL(s) refers to output by an independent that does notcompete with R, qLR(s) refers to output by an independent that does compete with Rand qR(s) refers to output by one of R’s stores.

We then take the expressions [A2] and [A3] and substitute these values into [3] togive an expression for spot market demand, QS(s).

[A5] QS(s) = {(m–r) n (a–s) + r (n–1) (a–2s)} / (n+1) b

Multiplying [A5] by s, equating sQS(s) with F and rearranging yields:

[A6] (m n + (n–2) r) s2 –a (m n–r) s + (n + 1) b F = 0

The equilibrium spot market price is the lowest root of [A6]. F is assumed lowenough such that the above quadratic has two roots for any value of r in the range 1 r m.

Apply the implicit function rule to [A6] to find ds/dr:

[A7] ds/dr = s (a + (n–2) s) / { a (m n–r) –2 (m n + (n–2) r) s}

From lemma 1, we know that ds/dr > 0 and hence the denominator of [A7] is positive:

[A8] a (m n–r) > 2 (m n + (n–2) r) s

Since M1’s marginal cost of production is zero, welfare, Wj in local market j is thearea under the demand curve in the range 0 to Qj. This equals:

[A9] Wj = ½ (a + Pj) Qj

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Let subscript R denote one of the r markets where R has a presence and subscript Ldenote one of the (m–r) markets where R does not have a presence:

[A10] PR = (a + (n–1) s) / (n + 1)

[A11] QR = (n a–(n–1) s) / (n + 1) b

[A12] PL = (a + n s) / (n + 1)

[A13] QL = (n a –n s) / (n + 1) b

Combining [A9] to [A13] total welfare, W can be written:

[A14] { a2 m n (n + 2) –2a (m n–r) s –(m n2–(2n–1)) s2 } –F_________________________________________________________________________________________________

{ 2 b (n + 1)2 }

Recall that s is a function of r and note that the denominator does not depend on r.This means that the sign of dW/dr is determined by the sign of the derivative of thenumerator with respect to r:

[A15] sign dW/dr = sign { 2 a s + (2n–1) s2 –2a (m n–r) ds/dr–2s (m n2–(2n–1)r) ds/dr }

Substituting [A7] for ds/dr and rearranging yields:

[A16] –s2 {a (m n (2n + 1)–3r) + 2 m n (n2–1) s }________________________________________________________________________________

{ a (m n–r) –2 (m n + (n–2) r) s }

From [A8] and noting that 1 r m and n 2, the sign of [A16] is negative. Thustotal welfare is decreasing in r. QED.

Lemma 3: When local demand P(Q) is concave, i.e. P’(Q) < 0 and P’’(Q) < 0, asufficient condition for welfare to decline as a result of R purchasing an additionalstore in a target market is: (n –1) dsb(r)/dr > sb(r)

Proof. First note that with concave demand and constant marginal costs, output in alocal market is decreasing in the sum of the marginal costs of the stores supplying thatmarket as we demonstrate below.

Given Cournot competition among n stores with constant marginal cost, i, facing amarket demand curve P(Q), the first order condition for store i (i = 1,2,..., n) yields:

[A17] P(Q)– P’(Q) qi–i,= 0.

Summing for all stores and rearranging yields:

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[A18] n P(Q) + P’(Q) Q –ii = 0

Since P’(Q) < 0 and P’’(Q) 0, by the implicit function rule, dQ/d(ii) < 0.

When R purchases an additional store, that store’s input price falls to zero from sb(r).The (n–1) other stores face a rise in their input price equal to dsb(r)/dr. If the lattereffect dominates the former then the sum of marginal costs increase and so output inthe target market falls. This is a sufficient condition for the merger to harm welfare.QED.

Lemma 4: d2W(r)/dr2 < 0. In other words, the more stores that R owns, the moreharmful (or less beneficial) is a further merger.

The proof has the following stages.

Stage 1: The expression

[A19] (n –1) dsb(r)/dr –sb(r)

is increasing in r.

Proof of stage 1. From [6] and [7], [A19], dsb(r)/dr is:

[A20] sb(r) {n qL(sb(r))–(n–1) qLR(sb(r))} / (1–(sb(r))) QS(sb(r))

Substituting [A20] into [A19] and rearranging (and dropping the arguments to avoidclutter) yields:

[A21] (1/m) (1–(n–1) qLR / n qL) { r + (n–1)/(1–)} –1

By inspection, [A21] is increasing in r. First consider qLR/qL. As r increases, so doessb. Both qLR and qL fall but qLR falls more quickly because output is more sensitive toa rise in the spot price in markets where independents compete with R as opposed tocompeting only with other independents. As qLR/qL falls the term in bold bracketsincreases.

Now consider terms in curly brackets. The first term is clearly increasing in r.Furthermore, as r increases, spot market demand becomes more elastic. Thus, increases and the expression (n–1)/(1–) goes up. QED.

Stage 2. We now complete the proof of lemma 4. Since [A19] is increasing in r, thismeans that as r goes up, the raising rivals’ cost effect becomes stronger. Thecontribution to welfare from the target market decreases (i.e. becomes less positive or,when [8] holds, becomes more negative). Further, the contribution to overall welfarefrom markets other than the target market is more negative (because output in thosemarkets declines by a greater amount). QED.

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Proposition 3: If local demand is of the form:

[A22] Pj(s) = a –b Qj(s) k

where k 1, then after a point (i.e. for r > r0, where 1 r0 < r m) a mergerharms welfare.

From [A22], Cournot output in a local market where R does not compete is:

[A23] QL(s) = { (n a–n s ) / (n + k) b } 1/k

and where R has a presence:

[A24] QR(s) = { (n a–(n–1)s ) / (n + k) b } 1/k

An independent that competes with R has output:

[A25] qLR = LR(s) QR(s)

where QR(s) is total output in that market and LR(s) is the independent’s market share:

[A25] LR(s)= (1/k) (k a–(k+1) s ) / (n a–(n–1) s ) < 1/n

Since R has a lower input price than any store with which it competes, it must be thecase that LR(s) < 1/n.

Suppose that R has no stores at all so that all purchases are made on the spot market.We can think of this as r = 0.30 In this case total output (i.e. summed across all mmarkets) is the same as spot market output:

[A26] m { (n a–n sr=0 ) / (n + k) b } 1/k

As before, assume that competition among suppliers means that the spot market priceis the lowest that allows one manufacturer to cover its fixed cost:

[A27] sr=0 m { (n a–n sr=0 ) / (n + k) b } 1/k = F

Now suppose that r = m so that R purchases as many stores as possible at stage 1 (wewould expect this as R’s profits are increasing in r). In this case spot market outputis:

[A28] m (n–1) LR(sr=m) { (n a–(n –1)sr=m ) / (n + k) b } 1/k

Thus:

30 Even though we have assumed r 1 in the main text nothing turns on this assumption. By setting r =0 we remove stage 1 of the game so R does not exist and there is no buyer power.

25

[A29] sr=m m (n–1) LR(sr=m) { (n a–(n –1)sr=m ) / (n + k) b } 1/k = F

Combining [A27] and [A29] we have:

[A30] sr=0 (n a–n sr=0 )1/k = sr=m (n–1) LR(sr=m) (n a–(n –1)sr=m ) 1/k

Note that if total output is the same when r = 0 and when r = m, it must be the casethat the sum of marginal costs in each local market is the same:

[A31] n sr=0 = (n–1) sr=m

However, if [A31] holds, the right hand side of [A30] is smaller than the left handside (from [A25]). To restore the balance we require that:

[A32] n sr=0 < (n–1) sr=m

since the left hand side of [A30] is increasing in sr=0. To see this take the naturallogarithm of the left hand side of [A30] and differentiate with respect to sr=0. Thisyields:

[A33] 1 / sr=0 –1 / k(a–sr=0 )

Thus the left hand side of [A30] is increasing in sr=0 if:

[A34] k a - (k + 1) sr=0 > 0

which surely holds from [A25].

From [A32] the sum of marginal costs is lower and hence output higher when r = 0than when r = m. Since all local markets are identical at these extremes, W(r=m) <W(r=0).

From lemma 4 we know that as r increases welfare either increases more slowly ordeclines more quickly (although we have focused on r in the range 1 r m, it isstraightforward to see that proposition 5 applies for 0 r m). Therefore, even ifwelfare initially increases in r, eventually we must reach some point, r0, wherewelfare starts to decrease with r to ensure that W(r=m) < W(r=0). QED.